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I 


THE  SYMMETRIC  FUNCTION  TABLES 
OF  THE  FIFTEENTHS 


INCLUDING 

AN  HISTORICAL  SUMMARY  OF  SYMMETRIC  FUNCTIONS  AS 
RELATING  TO  SYMMETRIC  FUNCTION  TABLES 


BY 

FLOYD  FISKE  DECKER 

ASSISTANT  PROFESSOR  OF  MATHEMATICS  AT  THE  SYRACUSE  UNIVERSITY 


WASHINGTON,  D.  C. 

Published  by  the  Carnegie  Institution  of  Washington 

' 

1910 


THE  SYMMETRIC  FUNCTION  TABLES 
OF  THE  FIFTEENTHS 


INCLUDING 


AN  HISTORICAL  SUMMARY  OF  SYMMETRIC  FUNCTIONS  AS 
RELATING  TO  SYMMETRIC  FUNCTION  TABLES 


BY 

FLOYD  FISKE  DECKER 

ASSISTANT  PBOFESSOR  OF  MATHEMATICS  AT  THE  SYRACUSE  UNIVERSITY 


WASHINGTON,  D.  C. 

Published  by  the  Carnegie  Institution  of  Washington 

1910 


1  IRfi&PV 

UNIVERSITY  OF  ILLINOIS 
AT  URBANA-CHAMP4I0N 


CABNEGIE  INSTITUTION  OF  WASHINGTON 
Publication  No.  120 


PRESS  OF  J.  B.  LIPPINCOTT  COMPANY 
PHILADELPHIA 


ve.c  t 


5"  I  $.8 A Ixffi 

H  353 


THE  SYMMETRIC  FUNCTION  TABLES 
OF  THE  FIFTEENTHIC. 


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cfc' 


My  attention  having  been  directed  by  Prof.  W.  H.  Metzler,  of  the 
Syracuse  University,  to  the  desirability  of  having  a  complete  table  of  the 
symmetric  functions  of  the  fifteenthic,  I  have  completed  the  work  here¬ 
with  presented.  Mistakes  may  have  escaped  the  scrutiny  of  the  computer, 
and  should  they  be  found  he  will  deem  it  a  favor  to  have  them  called  to 
his  attention.  A  brief  historical  summary  of  that  part  of  the  subject  of 
symmetric  functions  which  is  connected  with  the  computation  and  use  of 
the  tables  is  also  given.  Care  has  been  taken  to  bestow  credit  where  credit 
is  due. 


HISTORICAL  SKETCH. 


After  the  sixteenth  century  had  produced  solutions  of  the  cubic  and 
the  quartic  equations  and  had  unsuccessfully  attacked  the  solution  of  the 
quintic,  attention  gradually  turned  toward  the  relations  between  the  roots 
and  coefficients  of  equations.  In  fact,  about  the  middle  of  that  century, 
Yieta,  an  officer  of  the  French  Government,  observed  that  in  the  equation 


x2  +  a1x  +  a2  =  0 


if  ax  is  the  negative  of  the  sum  of  two  positive  numbers  oq,  a2,  of  which  a2 
is  the  product,  then  ax  and  a2  are  roots  of  this  equation.  He  failed  to 
notice,  however,  the  universality  of  the  relation,  as  he  recognized  positive 
roots  only. 

Knowledge  of  the  relations  between  the  roots  and  coefficients  of  equa¬ 
tions  grew  little  by  little  until,  at  the  time  of  Newton — when  negative  and 
imaginary  roots  had  gained  recognition — between  the  roots  a1,  a2,  a3,  ...  .  an 
and  the  coefficients  ax,  a2,  a3,  .  .  .  an  of  the  equation 


xn  +  a1xr'-1-\-  a2xn~2  H - +arxn~r  + - \-an  =  0 

the  following  relations  were  understood,  namely 

2a1  =  a1  +  a2  +  a3+ •  •  • +an  =—ax 

Saxa2  =  a1a2  +  a1a3+  •  •  •  +a2a3+  •  •  •  =  a2 

S axa2  .  .  .  aT  =  axa2  .  .  .  ar  +  axa2  .  .  .  ar_1ar+1+  •  •  •  =  (  —  l)rar 
axa2  .  .  .  an  =(-l)na„ 


(1) 


191800 

* 


4 


THE  SYMMETRIC  FUNCTION  TABLES  OF  THE  FIFTEENTHIC. 


A  function  of  several  quantities  which  is  not  changed  when  any  two 
of  the  quantities  are  interchanged,  such  as  Xaxa2,  is  called  a  symmetric 
function  of  the  quantities.  Such  a  function  of  the  roots  of  the  equation 
(1)  has  been  designated  by  inclosing  the  exponents  in  brackets,  expressing 
the  repetition  of  a  number  by  an  exponent :  thus 

2afafa|a4a5a6 


would  be  written  [322  l3]. 


In  particular,  a  symmetric  function  of  the  form 


[K]  =  a*  +  af  -f  •  •  •  •  +a 


K 

n 


is  called  an  elementary  symmetric  function. 

Newton  in  his  Cambridge  Lectures,  Arithmetica  Universalis  (published 
in  1707),  adds  the  following  relations  between  the  coefficients  and  the  ele¬ 
mentary  symmetric  functions  of  the  roots  of  equation  (1) 

[2]  =  2cx2  =  a?  A  A  •  •  •  +  a2  —  a?  —  2a2 

[3] =2a?  =  a?  +  a!+  •  •  •  +a3=  —  a?  +  -  3a3  _  ^ 

[4]  =  2a i  =  a4  -|-  0-2  A  ■  ■  ■  A  ot-n  =  —  4ct2&2  A  2n2  A  4cqci3  —  4 a4 

etc. 


These  results  were  utilized  for  the  calculation  of  resultants  of  pairs  of 
equations  as  early  as  the  middle  of  the  eighteenth  century  by  Euler  and 
Cramer,  who  were  interested  in  the  problem  of  the  number  of  intersections 
of  two  curves. 

Toward  the  close  of  the  eighteenth  century  Waring  published  a  for-  - 
mula1  giving  the  value  of  any  coefficient  in  the  expression  for  [K]  in  terms 
of  the  coefficients:  for  example,  the  coefficient  of  a4a|a2,  written  (412212) , 
in  [10]  is 


,  1v+a+»-10-(l  +  2  +  2-l)! 
^  ;  1 !  2!  2! 


or  —60 


(4) 


It  is  to  be  noticed  that  this  formula  is  a  generalization  of  Newton’s  for¬ 
mulas  (3). 

A  more  important  contribution  made  by  Waring  to  the  subject  was 
a  reduction  formula2  by  which  any  symmetric  function  [xYx2  .  .  .  xn]  could 
be  expressed  in  terms  of  elementary  symmetric  functions.  This  formula, 


1  In  general,  the  coefficient  of  a,Aia2A2  .  .  .  a,/n  in  [X],  as  given  by  Waring,  is 

where  J  =  b  +  •  '  +*» 

See  Waring:  Miscellanea  Analytica  de  Aequationibus  Algebraicis  et  Curvarum  Proprietatibus,  Cambridge, 
1762. 

2  The  following  is  Waring’s  statement  of  the  formula  (see  Waring:  Meditationes  Algebricae:  also  M. 
O.  Terquem:  Nouvelles  Annales,  1849).  Let 

*Sa=*ia+a:2a+  •  •  ‘  +xna 
Sb=X1b  +Xj>  +  ■  ■  ■  +Xnb 

St  =X/  +X2‘  +  ■  ■  ■  +xj 


HISTORICAL  SKETCH. 


5 


in  conjunction  with  his  formula  (4),  provided  a  method  for  expressing  any 
symmetric  function  of  the  roots  in  terms  of  the  coefficients;  but  the 
method  is  a  rather  cumbersome  one,  as  may  be  judged  from  the  following 
comparatively  simple  application,  the  calculation  of  S a\a.\al : 


A  =  $4 
and 


$*•$, 


B  =  A 


$4  +  3  $4  +  2  $ 

ry  rr  i  rr  n  T  n  rt 
.O4  *^3  O  4  *  ^2  ^3*^2- 


3  +  2 


Satalag 


l  =  A  -B  +  2C 

—  $4  •  $3  "  $2  $2  '  $4+3  " 

=  Saf  •  •  2a?  —  2a? 


p  _  a  r  $4+3+2  1 

^  o’  cr  rr 

LO4  *  |J3  *  OoJ 


$3  ’  $4+2  $4  '  $3  +  2  +  2*S, 


-’4+2 

2a? -2a? 


'4+3+2 

V„6_y„4.ya5  +  2Va9 


-’4 

ta? 


3+2 

at 


(5) 


During  the  very  year  (1771)  that  Waring  announced  his  formula  (5), 
Vandermonde  followed  it  by  actually  calculating  the  values  of  various  sym¬ 
metric  functions  of  the  roots  of  an  equation  in  terms  of  the  coefficients, 
publishing  the  results  in  the  form  of  tables. 

A  very  pretentious  work,3  with  the  harmless  title  “A  Collection  of 
Examples,  Formulae,  and  Exercises  on  the  Literal  Calculus  and  Algebra,” 
but  announcing  that  the  author,  Meyer  Hirsch,  had  “discovered  the  gen¬ 
eral  solution  of  equations,”  appeared  in  Berlin  in  1808.  In  this  work 
symmetric  functions,  their  calculation,  and  their  applications — including  a 
method  for  their  use  in  calculating  the  resultants  of  pairs  of  equations — - 
occupy  an  important  position;  and  in  an  appendix  are  to  be  found  the 
tables  themselves  up  to  and  including  the  tenthic  (that  is,  the  table  giving 
the  values  of  the  symmetric  functions  of  an  equation  of  the  tenth  degree) . 

In  his  tables  he  arranges  the  coefficients  according  to  the  dictionary 
method,  or  rather  according  to  the  reverse  of  the  dictionary  method.  For 
example,  for  the  tenthic  he  uses  the  equation  in  the  form 

xn  —  Axn~l  A  Bxn~2  —  •  •  •  — Ixn~9  A  Kxn~10  —  •  •  •  =0 


where  the  positive  integers  a,  b,  .  .  .  t,  are  all  different  and 


A=Sa-  Sb-  .  .  .St 
B 


S(i\b  ^  Sfi  1  c  ^ 


A  So- 
lSa-Sb  '  Sa-S, 
1  r  *s<x+&+ 

~  lSa-Sb- 

Sa+b+c+d 


|  Sb+c  | 


Sb-Sc 


c 

D  =  A 


*^a+6+c  -  ^a+b+d  , 

s7  +  AAsV-Sa 


Sa-Sb-Sc-Sd 


+ 


n  r>  \  r  Sa+b  *  Sc  ■  :i 

BB=A[sa.sb-scssd  +  ---\ 

■pi _  4  j _ *S>a+fr+c+<i+e 

~[s~-sb-s^d-s< 

L  * 

f-a[s7- 


Sb-Sc-Sd-Se 

$a+b+c+d+e+/ 


+ 

+ 


Sb  ■  Sc  ■  Sa  ■Se-SJ 


Then 

Zx«x2b  .  .  .  xn‘=A-B  + 1  •  2C— 1  •  2-  3D+1  •  2-  3  -4E+1-  1  •  BB 

-11-  2BC—1  -2-3  •  4-  5E+1  •  2-  3-4  -  5-6G+1  ■  1  •  2  -  3 BD 
-1  •  1  ■  2  -  3-4RE  +  1  •  2  -  1  •  2CC-1  -21  ■  2-3CH-1-  1  •  1 BBB 
+  1-1-1-2 BBC+-  •  ■ 

Each  letter  introduces  into  a  coefficient  which  it  enters  a  factor  as  follows: 


B,—  l  C,  1-2  D, -1-2-3  E,  1-2-3-4  F,- 1  •  2  •  3  •  4  •  5;  etc. 

When  any  of  the  exponents  become  equal,  it  is  necessary  to  divide  the  quantities  A,  B,  C,  D,  etc.  by  the 
factorials  of  the  appropriate  numbers.  This  theorem  was  independently  discovered  by  Gauss  in  1816. 
3English  Translation  by  Ross,  London,  1827. 


6 


THE  SYMMETRIC  FUNCTION  TABLES  OF  THE  FIFTEENTHIC. 


and  his  arrangement  from  right  to  left  begins 

K  I A  HB  HA2  GC  GBA  GA3,  etc. 

He  arranges  the  functions  according  to  the  number  of  parts  in  their  par¬ 
titions,  that  is,  according  to  the  number  of  a’s  composing  the  individual 
terms  (3221,  which  when  inclosed  in  brackets  signifies  Safa^a|a4,  is  called 
a  four  part  partition  of  9) ,  beginning  with  the  least.  Thus  for  the  tenthic 
his  arrangement  begins  at  the  top  with 

[10]  [91]  [82]  [73]  [64]  [52]  [812]  [721],  etc. 

This  plan  gives  his  table  the  triangular  form  with  the  principal  diagonal 
elements  each  unity,  as  the  following  reproduction  of  his  fifthic  indicates. 


lO 

Qq 

CO 

<NJ 

O 

<N 

O 

oq 

kq 

[5] 

1 

-5 

+  5 

+  5 

-5 

-5 

+  5 

[14] 

1 

-3 

-1 

+  5 

-1 

-5 

[23] 

1 

-2 

-1 

+  5 

-5 

[123] 

1 

-2 

-1 

+  5 

[122] 

1 

-3 

+  5 

[132] 

1 

-5 

[l5] 

1 

A  half  century  later  Brioschi  announced  a  differential  equation,4  of 
use  in  calculating  symmetric  functions,  as  follows: 

Consider  equation  (1)  and  let  sK  =  \K~\  =  2af ;  then 


Its  application  may  be  illustrated  by  the  calculation  of  [321].  Write 


[321]  =  CxCIq  +  c2a5a1  +  c3a4a2  +  c4a4a2  +  c5aj  +  c(>a3a2ai  +  c7a%  I 


Next  express  [321]  in  terms  of  the  s’s  by  means  of  (5)  thus 

[321]  —  S3S2S1  SgS^  s4s2  S3  T  2sg  II 

then  apply  (6) ,  making  K  =  6 

— |c1  =  2,  or  c4  =  — 12 


4MvBrioschi:  Annales  de  Tortolini,  Rome,  1854. 


HISTORICAL  SKETCH. 


7 


then  making  K  =  5 

— i(c2«i  +  c1a1)  =  — >SX 

and  since  =  —  a4  c2  =  7, 

making  K  =  4 

—  t  (c3a2  +  c4af  +  c2a\  +  c^o)  =  -  S2  =  —  (a?  —  2  a2) 


and  by  equating  the  coefficients  of  the  a’s,  c3  +  cx  =  —  8  and  c2  +  c4  =  4,  giving 


c3  =  4  c4  =  —  1 

making  /v  =  3 

— |  [2c5a3  +  c6a4a2  +  ch  (c3a2  +  c4af)  +  a2c2a4  +  cla]  =  s4s2  —  2s3 

from  which  are  obtained 


c5  3  c6  —  1 
Finally  c7  is  found  to  be  0,  and 


[321]  =  —  12a6  +  7  abax  +  4a4a2  —  3a4al  —  3  a§  +  aza2a4 


By  the  use  of  a  determinant  and  symbolic  multiplication,  Brioschi 
also  expressed  Waring’s  reduction  formula  (5)  in  a  much  simpler  form 
than  did  its  discoverer;  thus 


where 

and  in  general 


^11^12  • 

•  •  Uln 

•  ..*»]  = 

R-21^22  • 

■  •  U2n 

(7) 

V'nlV'ni  • 

■  •  Unn 

'M'rsU8r  = 

[*S  +  *r] 

UrsUst  .  .  .  Uvr=  [xr  +  x8  +  •  •  •  +*,] 


About  the  same  time  (1857)  Cayley5  republished  the  Hirsch  tables, 
using  the  equation  (1)  and  reversing  the  order  of  the  coefficients  in  the 
tables,  so  that  Hirsch’s  principal  diagonal  became  his  sinister  diagonal. 
He  proved  that  each  of  its  elements  must  be  (  — 1)”’,  where  w,  called  the 
weight,  is  defined  for  [x4x2  .  .  .  x„]  by  the  equation 


W  =  x1  +  x2+  ■  •  •  +xn  (8) 

for  example  the  weight  of  1,a\a2az  is  7. 

Since  if  a\'a2-  .  .  .  a„n,  written  (/fik2  .  .  .  k n)  is  to  have  a  coefficient 
other  than  0  in  the  expression  for  [x4x2 .  .  .  xn]  in  terms  of  the  coeffi¬ 
cients,  the  relation 


^1  +  ^2+  ‘  '  ‘  +  Xn  —  ^1  +  2k2  +  •  •  • 

must  obtain,  we  may  also  calculate  the  weight  from  the  equation 

w  =  / l1  +  2k2+  •  •  •  +  ukn 


5  A.  Cayley:  Philosophical  Transactions  of  the  Royal  Society  of  London,  vol.  147,  1857. 


8  THE  SYMMETRIC  FUNCTION  TABLES  OF  THE  FIFTEENTHIC. 

Cayley  used  the  term  conjugate  partition,  due  to  Ferrers,  the  meaning 
of  which  may  be  shown  by  the  following  example  exhibiting  its  calcula¬ 
tion.  To  get  the  partition  conjugate  to  7,  3,  22  a  row 
of  seven  dots  is  written;  under  these,  beginning  at  the 
left,  a  row  of  three,  under  these  two  rows  of  two  dots 
each;  then  the  dots  in  the  columns  are  counted  and  .  . 

found  to  be  4,  4,  2,  1,  1,  1,  1  respectively,  and  accord-  4;  4)  2,  1,  1,  1,  1 

ingly  42,  2,  l4  is  called  the  conjugate  partition  of  7,  3,  22. 

In  general  the  partition  conjugate  to  xu  x2,  .  .  .  xn  is 

l*1-*2,  2K2~K3,  ■  •  •  (ft—  ft*n 

Cayley’s  theorem  in  regard  to  the  sinister  diagonal  elements  may  now 
be  given  the  form, 

coefficient  of  (P)  in  [Q]  is  (  —  1)“  (9) 

if  Q  is  the  partition  conjugate  to  P. 

Cayley  noticed  a  symmetry  in  the  constituents  of  the  tables,  observing 

that 

coefficient  of  (P)  in  [P]  is  same  as  the  coefficient  of  (P)  in  [P]  (10) 

In  the  following  year  the  Italian  Betti6  independently  observed  this 
symmetry  and  showed  the  necessity  for  it,  since  which  time  it  has  been 
known  as  the  Cayley-Betti  law  of  symmetry. 

Cayley  reduced  the  number  of  coefficients  to  be  calculated  in  a  table 
by  observing  that  ak  would  not  occur  in  any  symmetric  function  involving 
less  than  k  numbers  in  its  partition:  for  example,  in  the  fifthic  a4  occurs  in 
[132]  and  [l5]  only.  He  observed  that  it  is  possible  to  keep  the  sinister 
diagonal  elements  (  —  1)™  by  arranging  the  partitions  representing  the  sym¬ 
metric  functions  in  the  same  order  as  those  representing  the  coefficients, 
giving  the  table  a  triangular  form.  He  saw  also  that  the  table  could  be 
made  symmetrical  by  arranging  the  self-conjugate  partitions  (such  as  433) 
at  the  middle,  in  one  order  for  the  functions  and  in  the  reverse  for  the 
coefficients;  but  that  the  table  could  not  at  once  possess  both  properties. 
He  expressed  his  preference  for  the  latter  method. 

Cayley  called  attention  to  the  ease  with  which  resultants  could  be 
calculated  if  the  appropriate  symmetric  function  tables  were  at  hand. 
He  corrected  the  score  of  mistakes  in  the  Hirsch  tables,  making  them 
reliable  for  computation.  They  may  be  found  in  Salmon’s  “  Modern 
Higher  Algebra.” 

Faa  Di  Bruno,  who  a  little  later  took  up  the  work,  had  a  formula 
giving  multipliers  for  the  constituents  in  a  row  or  column  such  that  the 
sum  of  the  products  (except  in  the  case  of  the  row  or  column  with  the 


9M.  Betti:  Annales  de  Tortolini,  Rome,  1858. 


HISTORICAL  SKETCH. 


9 


partition  lfc)  would  be  0.  For  example,  the  coefficient  of  a3a2  in  a  function 
of  the  fifthic  would  require  the  multiplier 

gf*!  or  10  (11) 

and  the  complete  check  for  [41]  would  be 

5- 1-1  -5-5-  10  +  1  -20  +  3 -30-1 -60  +  0- 120  =  0 


In  general  the  coefficient  of  .  .  .  Xn)  in  the  ic-thic  would  require 

the  multiplier 


w\ 


(12) 


He  published  in  1875  the  first  ten  symmetric  function  tables,  adding 
to  the  list  the  eleventhic.  They7  may  be  found  in  his  “  Theorie  der 
Binaren  Formen.”  In  them  certain  misprints  may  be  noted:  the  values 

of  the  coefficients  where  the  misprints  occur  are  here  given, 

ing  the  coefficient  of  (P)  in  [Q]. 


Eighthic : 


Q 


signify- 


Ninthic : 


7 They  may  also  be  found  in  the  Gottingen  Naehrieht  (1875). 


10 


THE  SYMMETRIC  FUNCTION  TABLES  OF  THE  FIFTEENTHIC. 


Tenthic — continued : 


In  1881  a  formula  giving  the  number  of  symmetric  functions  of  a 
given  weight  was  announced  by  Forsyth.8  An  application  of  it,  to  get 
an  idea  of  the  lengths  of  tables  not  then  computed,  gives  the  number  of 
symmetric  functions  of  weight  15  to  be  176,  of  weight  22  to  be  1001,  and 
of  weight  30  to  be  5595. 

Hammond,  in  the  Proceedings  of  the  London  Mathematical  Society 
for  1881-82  (vol.  13),  gives  a  convenient  method  for  calculating  those 
terms  in  [xxz2  .  .  .  *n]  which  contain  no  a  with  a  subscript  greater  than 
a  given  number  by  identifying  the  coefficients  with  coefficients  in  tables 
of  lower  weight.  Thus 

[54321]  =a5 

X  (terms  in  [4321]  containing  no  a  with  a  subscript  greater  than  5) 

+  terms  containing  a’s  with  subscripts  greater  than  5: 


that  is 

[54321]  =  a5X 

[  (432 1 )  —  3  (432)  —  3  (421 2)  -f  4  (422)  —  3  (522 1 )  +  4  (53 1 2)  +  5  (532)  —  5  (52)  ] 
+  terms  containing  a’s  with  subscripts  greater  than  5. 


He  also  gives  a  formula  for  the  computation  of  symmetric  functions 
by  the  use  of  auxiliary  functions. 


A.  R.  Forsyth:  Messenger  of  Mathematics,  vol.  10,  1880-81. 


HISTORICAL  SKETCH. 


11 


In  the  same  year  Rehorovsky9  published  the  eleventhic  and  the 
twelfthic,  with  the  aid  of  an  additional  formula  giving  the  sum  of  the 
constituents  in  a  row  or  column;  for  example,  it  gives  the  sum  of  the 
constituents  in  32221  as 


(  1N2+2+l(2  +  2+l)  ! 

^  ’  2 !  2 !  1 ! 


-30 


(14) 


In  the  same  year  Durfee  showed  that  the  scheme  of  arrangement  in  a 
triangle  is  always  possible  in  two  ways,  and  in  the  American  Journal  of 
Mathematics  he  published  the  twelfthic  arranged  in  one  of  them,  checking 
his  results  by  the  Cayley-Betti  law  of  symmetry  (10)  as  well  as  by  the  law 
for  the  sum  of  the  constituents  in  a  row  or  column  (14). 

Major  MacMahon,  in  his  article  in  The  Proceedings  of  the  London 
Mathematical  Society  for  1883-84  (vol.  15),  gives  several  symmetric  func¬ 
tion  formulas,  four  of  which  will  be  here  noted.  The  first  formula  enables 
one  to  write  down  all  those  terms  of  the  highest  degree  in  the  n’s  of  a  sym¬ 
metric  function  from  a  function  in  a  table  of  lower  weight.  For  example: 


[3222]  =  a\a2  —  2  aba3a2  —  a5a4a1  +  5a|  +  2  a6a|  -(-  3a6a3aL  —  9cc6a;4 
—7a7a2al  +  6a7a3  +  7a8a\  +  a8a2  —  lSagftj  +  15a10 


Therefore 


[3322]  =  (  —  1) 13-10  (a5«3  —  2a6a4a3  —  a6a5a2  +  +  2a7a|  +  3a7a4a2 

—  §a7abax  —  7a8a3a2  +  §a8aial  +  7cc9a2  +  —  \bawa2al  +  15ana2) 

+  terms  of  lower  degrees. 


(15) 


Another  of  his  formulas  enables  us  to  compute  any  function  from  a 
single  function  of  lower  weight  by  means  of  a  differential  operator  V _r. 
In  particular 


F_1  =  cr1 


d 


da0 


+  2a2 


d 

da7 


+  3c3 


d 


dan 


+ 


and  by  means  of  his  equation 

V-rty]  =  (  —  l)Kx[_xlr]  (16) 

we  may  calculate,  for  example,  [41]  from  [4].  We  first  write  in  the 
homogeneous  form 

[4]  =  —  4a4«o  +  4a3a;1ao  +  2a|«o  —  4a2a2a0  +  cc\ 

Then 


[41]  =  —\V  _l[4] 

=  —  7  [  —  12a4a1ao  +  8a3ala0  +  4 ala 7a0  —  4  a2a\  +  8a3a2al  —  1 
+  8a2a\  4-  I2a3a2al  — 12  a3a\a0  +  16a4a,rto  —  20a,5a(]  ] 


or 


[41]  =  5  a5  —  a4a!  —  5a3a2  +  a3a\  +  3ala7  —  a2a^ 


•Rehorovsky:  Wien  Denkschriften  der  Kaiserliche  Akademie  der  Wissenschaften  in  Wien  Mathe- 
matisch-Naturwissenschaft-liche  Classe,  vol.  46,  Vienna,  1882. 


12 


THE  SYMMETRIC  FUNCTION  TABLES  OF  THE  FIFTEENTHIC. 


A  third  formula,  based  on  a  system  of  operators  developed  by  Ham¬ 
mond  in  The  Proceedings  of  the  London  Mathematical  Society  for  1882-83 
(vol.  14),  provides  a  method  for  calculating  any  symmetric  function  of  the 
n-thic  from  the  elementary  symmetric  functions  of  weight  n  and  lower,  l—l 
checks  being  provided  for  a  coefficient  of  a  term  containing  l  a’ s.  The 
operators  are  unusual  in  that,  while  calculation  by  means  of  them  is  very 
simple,  they  themselves  are  functions  of  elementary  symmetric  functions. 
One  application  of  them  to  the  calculation  of  symmetric  is  given  by 
Hammond  in  his  paper,  his  illustration  being  the  calculation  of  [341] 
from  [34]. 

A  fourth  formula  in  MacMahon’s  article  provides  a  check  by  giving 
the  sum  of  the  coefficients  of  the  terms  of  a  given  degree  in  a  symmetric 
function,  or  the  sum  of  the  coefficients  of  a  given  term  in  all  the  symmetric 
functions  with  a  given  number  of  parts  in  their  partitions.  He  first 
introduces  a  generalized  definition  of  weight 

1TV  — ri/ly_i +  r2kv  +  r3A,y+i+  •  •  • 

where 

rx=K  +  }K(K-l)(r-2) 


that  is,  where  rK  is  the  kth  of  the  r-gonal  numbers;  a  definition  which, 
it  is  to  be  observed,  makes  w2  =  W.  The  formula  for  the  sum  of  the 
coefficients  of  the  terms  of  the  mth  degree  in  the  a’ s  in  the  value  of 
[xf’xf2  .  .  .  x*n~]  (or  the  sum  of  the  coefficients  of  (xf'xf2  .  .  .  x„n)  in  all 
of  the  symmetric  functions  with  m-part  partitions)  is 


where 


_  )K+m-AK-l)\Wm+l 
1  Kx\K2\...Kn\ 

K=K1+K2+-  --+Kn 


(17) 


For  example,  the  sum  of  the  coefficients  of  terms  of  the  fourth  degree  in 
the  a’s  in  the  value  of  [S^1!3]  is 


(  n«+4-i(5-l)  1(5-1) 

1  ’  3! 1 !  1 ! 


20 


since 


Kx  =  3  K2  =  1  K3  =  0  K,  =  0  K5  =  l  K6  =  0  K7  =  0  .  .  .  Zn  =  0 

K  =  5  M  =  4  TFS  =  5g  •  1  =  5 


In  the  following  year  the  same  writer  published  an  article  in  the 
American  Journal  of  Mathematics  in  which  he  repeated  formula  (17)  and 
included  the  additional  check  that  any  function  except  [1A]  would  vanish 
if  for  each  coefficient  the  reciprocal  of  the  factorial  of  its  suffix  were  sub¬ 
stituted,  (18).  Accompanying  the  article  is  the  thirteenths,  arranged  like 
Durfee’s  twelfthic  and  checked  by  the  CaylejMBetti  law  of  symmetry,  (10), 
as  well  as  by  his  own  checks,  (17),  (18). 


HISTORICAL  SKETCH. 


13 


A  misprint  may  be  noted  in 


It  should  read 


=  1. 


In  1887  Durfee  published  in  the  American  Journal  of  Mathematics 
the  fourteenthic  arranged  according  to  his  second  (dictionary)  plan  and 
checked  not  only  as  was  his  twelfthic,  but  also  by  MacMahon’s  formula  (17) . 

In  1898  a  formula10  was  published  by  Metzler  which,  when  the  value  in 
terms  of  the  a’s  of  say  ^a\ala\a°4  is  given,  makes  it  possible  directly  to  write 
all  those  terms  in  say  2a?_0at_1a i~2a\~z,  involving  only  the  coefficients 

di,  d2,  d3,  d4. 

s  From  the  equation  for  ^,alalala4  in  the  homogeneous  form 


i a\a\a\u. 4  =  d3d2dxd  0  —  3d%d%  —  3d4d\d0  +  4a4a.2a0  +  7d5dxdl  —  12a6tto 


is  derived  the  equation 


V  4-0  4-1  4 
CL  o  rt- 


Z  a 


^9 


~2a\  3  =  d4_3d4_2d4. 


1&4-0  3d4-3d4. 


_4cq  _  icq  _q  - l-  4n4_4n4_2n4_ o 
+  terms  involving  d5,  d6,  etc. 


giving  in  the  nonhomogeneous  form 

^a\alala4  =  d4d3d2d4  —  3d\d\  —  3cc4a|  +  4a4cc2 

+  terms  involving  d5,  dG,  etc. 


(19) 


During  the  year  1899  there  appeared  in  the  American  Mathematical 
Monthly  a  series  of  articles  on  symmetric  functions  by  Roe.  Not  content 
with  Cayley’s  remark  as  to  the  ease  with  which  resultants  could  be  calcu¬ 
lated  if  the  appropriate  symmetric  functions  were  at  hand,  he  carried  the 
matter  farther,  identifying  the  problems  of  calculating  all  resultants  and 
calculating  all  symmetric  functions.  From  a  consideration  of  the  former 
problem  he  derives  a  complete  set  of  formulas  for  the  latter.  He  arranges 
them  in  three  classes  (fundamental,  reduction,  and  normal)  and  expresses 
them  by  means  of  a  new  symbol 

(*) 


which  represents  the  coefficient  of  (P)  in  [P] . 

His  fundamental  relations  consist  of  the  Cayley-Betti  law  of  symmetry 
(10)  and  Metzler’s  formula,  (19).  His  reduction  formulas  consist  of  four 
which  he  designates  as  reduction  (or  derivation)  formulas  and  one  which 
he  terms  the  formula  for  the  completely  reducible  form.  Of  the  four 
reduction  formulas,  one  is  Hammond’s  formula,  another  provides  for  the 
calculation  of  [lx] ,  the  third  for  the  reduction  of 

d%  [x4x2  .  .  .  xn] 


10  W.  H.  Metzler:  Proceedings  of  the  London  Mathematical  Society,  vol.  28,  1897-8. 


14 


THE  SYMMETRIC  FUNCTION  TABLES  OF  THE  FIFTEENTHIC. 


where  m  is  greater  than  xx,  while  the  fourth  states  that  a  function  of  the 
roots  of  an  equation  of  the  nth  degree  has  the  same  coefficient  of  the  term 
involving  (X^  .  .  .  Xr-1)  as  the  same  function  of  the  roots  of  an  equation 
of  the  (r—  l)th  degr'ee.  His  completely  reducible  form  is 


© 

t 

where P  is  the  partition  conjugate  to  Q,  the  formula  giving  Cayley’s  result  (9) . 

Roe’s  normal  forms  are  those  for  which  his  fundamental  and  reduction 
formulas  do  not  provide  a  reduction.  The  formula  is 


+1 


(T"“lh  .  .  .  n/n 
yn©(ra— l)^  .  .  . 


r=n 
.  V 

r=  1 


(1  +  Mr)  X 


0+  +  1  Ri  .  .  .  (r —  1)X-1+,?—  l(f-j-  1)^+1  .  t  _  \ 

1(m—  l)!“i  .  .  .  (m— r  —  l)^-i(m  —  —  r  +  !)/“>•+ 1  .  .  .  Oot/_ 


Thus  he  reduces  his  normal  coefficients  to  the  sums  of  coefficients  of  tables 
of  lower  weight.  For  example 


R)2134° 

,  403  , 


/0213\  / 023N 

R440V  (1  +  0)\30h 


(1+0)1 


'0T 

HO2. 


(—3+1=4 


that  is, 


coefficient  of  a3al  in  2a?  =  —  (1  +0)  X  coefficient  of  a3  in  2a?  —  (1+0) 
X  coefficient  of  cq  in  2ax  =  3+1=4. 


Roe  gives  also  a  formula  for  calculating  the  constituents  in  each  of  the 
first  four  lines  (or  columns).  Those  for  the  first  two  follow: 


First  line 


w(?l  1  +  ^-2+  •  •  •  +  Xn— 1) 


!) 


xfx2! . 


aj 


(20) 


where  ra  =  X1  +  X2  +  •  •  •  +  Xn. 


Second  line 


(po^i 


n 


(w-l)10.  .  .0, 


1 J/ (w  —  1)  (/C+X2+  •  •  •  2) ! 

1-1  ^  -l)ra,!...JL! 

+  +n—  1) 


w(x  i  +X2  + 


Xi !  X2 


V 


(21) 


where  h  =  2  when  m  =  2  and  h  =  1  when  m  >  2. 

Roe’s  formulas  were  used  by  the  writer  in  the  calculation  of  the 
fifteenthic  here  appended.  The  work  has  been  verified  by  the  use  of  the 
law  for  the  sum  of  the  coefficients  in  a  row  or  column,  (14) ,  and  by  Roe’s 
formulas,  (20)  and  (21). 


ILLUSTRATIONS  OF  USES  OF  SYMMETRIC  FUNCTION  TABLES. 


15 


ILLUSTRATIONS  OF  USES  OF  SYMMETRIC  FUNCTION  TABLES. 


To  illustrate  a  use  of  symmetric  functions  we  may  solve  the  quartic 
equation 

x4—x3  —  8x2  +  2x  +  12  =  0  (1) 

with  roots  say  a1}  a2,  a3,  a4.  We  may  begin  by  forming  the  cubic  resolvent 
in  Z,  where  Z  has  the  three  values  of  the  function 


The  equation  is 

akah  +  ai3au 11 

Z3-Z2Xax 

a2  +  Z2  a\a2a3  —  (Sa?a2a3a4  +  2ta?a|a|)  =  0 

(2) 

By  the  use  of  tables  of  symmetric  functions  this  may  be  written 

Z3  +  a2Z2  +  (axa3  —  4  a4)  Z  +  4a2cc4  —  aj  —  ct?a4  —  0 

(3) 

for  the  general  quartic 

X 4  +  axx3  +  a2x2  +  a3x  +  a4  =  0 

(4) 

or  for  (1) 

Z3  +  8Z2  —  50Z  —  400  =  0 

(5) 

Solving  the  cubic  (5) 

4  =  a4a2  +  a3a4  =  —8 

(6) 

@2  ~  ala3  +  a2a4  “  5^/  2 

(7) 

(33  -  oqoq  +  a2a3  =  —  5yj  2 

(8) 

and  since 

ax  +  a2-f  a3  +  a4=  1 

(9) 

we  may  solve  the  set  (6) ,  (7) ,  (8) ,  (9) ,  for  the  a’s  and  get 

oq  —  3  a2=  —  2  a3  =  yj  2  a4  =  —  -yj  2 

To  illustrate,  in  particular,  the  use  of  the  fifteenthic,  let  us  find  the 
resultant  of  a  cubic  equation  and  a  quintic  equation,  say 

f1(x)=x3-Sx2-2x+l  =  0  (10) 

and 

<pi(x)  =x5  —  2x  +  3  =  0  (11) 

For  the  resultant  of  the  cubic  equation 

f(x)  =  x3  +  axx2  +  a2x  +  a 3  =  0,  with  roots  ax,  a2,  a3  (12) 

and  the  quintic  equation 

$  (x)  ==  x5  +  bxx4  +  b2x3  +  b3x2  +  b^x  +  b5  =  0 

we  may  write 


=  -qM  -<pM 


(13) 

(14) 


11  To  see  that  +  a;3aj4  is  a  three-valued  function  one  may  try  various  permutations  of  the  sub¬ 
scripts  and  find  that  each  leads  to  one  of  the  forms  6,  7,  8. 


16 


THE  SYMMETRIC  FUNCTION  TABLES  OF  THE  FIFTEENTHIC. 


or  by  expanding 


5 


3 


(15) 


which  gives 


(16) 


By  the  use  of  symmetric  function  tables,  noting  that  aK  =  0  when  K>  3,  we 
may  write 

=  <4 12  —  2  ( -  4a|a2  —  2a|a?  +  4aia|a!  —  a3a$) 


+  3  ( —  5a|a!  +  5aj$a|  4-  5a|a2a?  —  ba3a2ax  +  a2) 

+  4  ( —  4a§a!  —  2  a3a22  +  4:a3a2a\  —  a3a\) 

—  6  ( —  3a|  +  7a3a2ax  —  a3a\  +  2a2  —  4  a\a\  4-  a2af) 
4-  9  (5 a3a2  —  5 a3a\  —  5a2ax  4-  5 a2a\  —  af) 

4"  8cl3  4-  12a2  +  18cii  -1-  27 


and  since  =  —3,  a2  =  —2,  and  cr3=l, 


RMl  =  2619 


12  To  read  the  value  of  Saja|a|  from  the  fifteenthic,  we  use  the  numbers  in  the  row  designated  53 
for  coefficients.  The  first  coefficient  is  in  column  headed  15  and  therefore  signifies— 5a16.  Thus  we  have 
2aja|a|=  —5a]5  +  5auax  +  5o,3a2  +  5a12a3  +  etc. 


ill 


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y  S: 


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MM  i  i  ! 


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18.4.2.1' 


UNIVERSITY  OF  ILLINOIS-URBANA 

Q.512.94D35S  C001 

THE  SYMMETRIC  FUNCTION  TABLES  OF  THE  FIF 


